##
**Mesh independence for nonlinear least squares problems with norm constraints.**
*(English)*
Zbl 0771.65030

This paper considers application of the Gauss-Newton method to discretizations (based on Galerkin approximations) of infinite dimensional nonlinear least squares problems of the form
\[
\min\| F(x)\|^ 2_ Y\quad\text{ under the constraint } \| x\|_ X\leq R.
\]
\(F\) is a sufficiently smooth weakly continuous function acting between the Hilbert spaces \(X\), the parameter space, and \(Y\), the output space. The analysis in the paper is local and assumes that a good estimate of the solution is available. The proposed variation applies the Gauss-Newton method to \(F(x)\) linearised round the current approximation \(x_ k\), but with the constraint unchanged.

A common observation is the stable behaviour for varying discretizations, and the constancy of the number of iterations to attain given terminating criteria. These phenomena are analysed and the mesh-independence for the constrained least squares problem is proved. In addition, sufficient conditions on the mesh independence are related to conditions which guarantee convergence of the Gauss-Newton method applied to the given problem. The paper concludes by illustrating the results with a numerical example, a two-point boundary value problem.

A common observation is the stable behaviour for varying discretizations, and the constancy of the number of iterations to attain given terminating criteria. These phenomena are analysed and the mesh-independence for the constrained least squares problem is proved. In addition, sufficient conditions on the mesh independence are related to conditions which guarantee convergence of the Gauss-Newton method applied to the given problem. The paper concludes by illustrating the results with a numerical example, a two-point boundary value problem.

Reviewer: A.Swift (Palmerston North)

### MSC:

65K05 | Numerical mathematical programming methods |

65J15 | Numerical solutions to equations with nonlinear operators |

90C48 | Programming in abstract spaces |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

90C30 | Nonlinear programming |